Quantization of channel state information in multiple antenna systems

ABSTRACT

A method of transmission over multiple wireless channels in a multiple antenna system includes storing channel modulation matrices at a transmitter; receiving quantized channel state information at the transmitter from plural receivers; selecting a transmission modulation matrix using the quantized channel state information from the stored channel modulation matrices; and transmitting over the multiple channels to the plural receivers using the selected transmission modulation matrix. In another embodiment, the method includes storing, at one or more receivers, indexes of modulation matrices generated by a capacity enhancing algorithm; upon a selected one of the one or more receivers receiving a transmission from the transmitter, the selected receiver selecting a modulation matrix from the stored modulation matrices that optimizes transmission between the transmitter and the selected receiver; the selected receiver sending an index representing the selected modulation matrix; and receiving the index at the transmitter from the selected receiver.

CROSS-REFERENCES TO RELATED APPLICATIONS

This application is a continuation of U.S. application Ser. No. 17/063,213 filed Oct. 5, 2020, which is a continuation of 16/436,532, filed Jun. 10, 2019, which is a continuation of U.S. application Ser. No. 14/628,570, filed Feb. 23, 2015, which issued as U.S. Pat. No. 10,320,453 on Jun. 11, 2019, which is a continuation of U.S. application Ser. No. 13/289,957, filed Nov. 4, 2011, which issued as U.S. Pat. No. 8,971,467 on Mar. 3, 2015, which is a division of U.S. application Ser. No. 11/754,965, filed May 29, 2007, which issued as U.S. Pat. No. 8,116,391 on Feb. 14, 2012, which claims the benefit of U.S. Provisional Patent Application No. 60/808,806, filed May 26, 2006, the entire disclosure of which is incorporated herein by reference.

BACKGROUND

The development of the modern Internet-based data communication systems and ever increasing demand for bandwidth have spurred an unprecedented progress in development of high capacity wireless systems. The major trends in such systems design are the use of multiple antennas to provide capacity gains on fading channels and orthogonal frequency division multiplexing (OFDM) to facilitate the utilization of these capacity gains on rich scattering frequency-selective channels. Since the end of the last decade, there has been an explosion of interest in multiple-input multiple-output systems (MIMO) and a lot of research work has been devoted to their performance limits and methods to achieve them.

One of the fundamental issued in multiple antenna systems is the availability of the channel state information at transmitter and receiver. While it is usually assumed that the perfect channel state information (CSI) is available at the receiver, the transmitter may have perfect, partial or no CSI. In case of the single user systems, the perfect CSI at the transmitter (CSIT) allows for use of a spatial water-filling approach to achieve maximum capacity. In case of multi-user broadcast channels (the downlink), the capacity is maximized by using the so called dirty paper coding, which also depends on the availability of perfect CSIT. Such systems are usually refereed to as closed-loop as opposed to open-loop systems where there is no feedback from the receiver and the transmitter typically uses equal-power division between the antennas.

In practice, the CSI should be quantized to minimize feedback rate while providing.satisfactory.performance of the system. The problem has attracted attention of the scientific community and papers provided solutions for beam-forming on flat-fading MIMO channels where the diversity gain is the main focus. Moreover, some authors dealt with frequency-selective channels and OFDM modulation although also those papers were mainly devoted to beamforming approach.

Unfortunately, availability of full CSIT is unrealistic due to the feedback delay and noise, channel estimation errors and limited feedback bandwidth, which forces CSI to be quantized at the receiver to minimize feedback rate. The problem has attracted attention of the scientific community and papers have provided solutions for single-user beamforming on flat-fading MIMO channels, where the diversity gain is the main focus. More recently, CSI quantization results were shown for multi-user zero-forcing algorithms by Jindal.

SUMMARY

This summary is provided to introduce a selection of concepts in a simplified form that are further described below in the Detailed Description. This summary is not intended to identify key features of the claimed subject matter, nor is it intended to be used as an aid in determining the scope of the claimed subject matter.

We present a simple, flexible algorithm that is constructed with multiplexing approach to MIMO transmission, i.e., where the channel is used to transmit multiple data streams. We use a vector quantizer approach to construct code-books of water-filling covariance matrices which can be used in a wide variety of system configurations and on frequency selective channels. Moreover, we propose a solution which reduces the required average feedback rate by transmitting the indexes of only those covariance matrices which provide higher instantaneous capacity than the equal power allocation.

DESCRIPTION OF THE DRAWINGS

The foregoing aspects and many of the attendant advantages of this invention will become more readily appreciated as the same become better understood by reference to the following detailed description, when taken in conjunction with the accompanying drawings, wherein:

Embodiments will now be described with reference to the figures, in which like reference characters denote like elements, by way of example, and in which:

FIG. 1 shows a typical transmission system;

FIG. 2 is a flow diagram showing basic system operation;

FIG. 3 is a flow diagram showing use of quantized mode gain in the system of FIG. 1 ;

FIG. 4 a is a flow diagram showing design of optimized orthogonal mode matrices for use in the system of FIG. 1 for single user case;

FIG. 4 b is a flow diagram showing design of optimized orthogonal modes for use in the system of FIG. 1 for a multi user system;

FIG. 5 is a flow diagram showing an embodiment of calculation of throughput at the transmitter in the system of FIG. 1 for single user case;

FIG. 6 is a flow diagram showing an embodiment of calculation of throughput at the transmitter in the system of FIG. 1 for multiple user case;

FIG. 7 is a flow diagram showing design of stored modulation matrices at the transmitter of FIG. 1 for multiple user case;

FIG. 8 is a flow diagram showing nested quantization;

FIG. 9 is a flow diagram showing design of mode gain matrixes for use in the system of FIG. 1 ;

FIG. 10 is a flow diagram showing design of power allocation matrices for use at the transmitter of FIG. 1 for multiple user case;

FIGS. 11A and 11B show the distribution of {circumflex over (V)} and Ŝ respectively in vector codebooks for a 2×2 MIMO system and N_(V)=2 and N_(S)=1. The two orthogonal eigenvectors in each {circumflex over (V)} matrix are shown using the same line style. The X-axis corresponds to the real entries in the first rows of {circumflex over (V)} and the YZ-plane corresponds to the complex entries in the second rows of the matrices. In case of Ŝ, the entries are presented for different power levels P in a normalized form Ŝ/P. FIG. 11A shows {circumflex over (V)} for N_(V)=2 and FIG. 11B shows Ŝ for N_(s)=1

FIG. 12A shows ergodic system capacity C and FIG. 12B shows average feedback bit rate R_(b) for different granularities of channel state information at the transmitter on the flat fading channel. System capacity is shown in logarithmic scale to better illustrate the differences between curves.

FIG. 13A shows capacity loss and FIG. 13B shows average feedback bit rate R_(b) on the frequency selective channel.

FIG. 14 shows sum-rates of cooperative, zero-forcing DPC and linear systems with full CSIT and 10 users with identical receivers. ..

FIG. 15 is an example of multi-user interference caused by partial CSIT.

FIGS. 16A and 16B show an example of the nested quantization of eigenmodes. The thick lines symbolize centroids {circumflex over (v)}(i) of the respective regions v_(i). FIG. 16A shows a coarse 2-bit quantizer and FIG. 16B shows a precise 2-bit quantizer of v₃.

FIG. 17 shows linear system sum-rates with full CSIT and varying feedback bit-rate for K=2. n_(T)=2 and all users with n_(R)=2 antennas.

FIG. 18 shows linear system sum-rates with full CSIT and varying feedback bit-rate for K +10. n_(T)=2 and all users with n_(R)=2 antennas.

FIG. 19 shows linear system sum-rates with full CSIT and nested CSI quantization with varying eigenmode coherence time τ_(eig) (N_(v)). n_(T)=2, K=10, n_(R)=2, N_(s)=1 and N_(v)=2,4.

DETAILED DESCRIPTION

One of the fundamental issues in multiple antenna systems is the availability of the channel state information (CSI) at transmitter and receiver[24]. The perfect CSI at the transmitter (CSIT) enables the use of a spatial water-filling, dirty paper coding and simultaneous transmission to multiple users, allowing the systems to approach their maximum theoretical capacity: Such systems are usually referred to as closed-loop as opposed to open-loop systems where there is no feedback from the receiver. Closed loop systems enable major increases of system capacities, allowing the operators to multiply their revenue and maintain high quality of service at the same time.

In this work, we describe a flexible approach to CSI encoding, which can be used to construct the linear modulation matrices for both single-user and multi-user networks. In both cases, the modulation matrices are composed of two independent parts: the eigenmode matrix and the diagonal power division matrix with the sum of entries on the diagonal equal to 1. The system operates as follows:

-   -   1. The receiver(s)[24] estimate(s) the respective multiple         antenna channel(s).     -   2. Each estimated channel is decomposed using the singular value         decomposition (SVD) to form the matrix of cigenmodes[30] and         their respective singular values[304].     -   3. If the system works in the single-user mode, all entries in         the codebook of transmitter eigenmode modulation matrices[32]         and all entries in the codebook of transmitter power division         matrices[32A] are tested at the receiver to choose their         combination providing highest instantaneous capacity. The         indices of the best transmitter eigenmode and power division         matrices are then sent[34].[34A] back to the transmitter.     -   4. If the system works in the multi-user mode, all entries in         the codebook of receiver eigenmode vectors[32] and all entries         in the codebook of receiver mode gains[32A] are tested at the         receiver[24] for best match with the estimated channel (the         matching function can be chosen freely by the system designer).         The indices of the best receiver eigenmode and power division         matrices[94] are then sent[34] back to the transmitter.     -   5. Based on the received[36],[36A] indices, the transmitter         chooses[38],[52],[62] the modulation matrix and uses it to         transmit[40] the information to one or more users[24] at a time.

Our proposed method allows to simplify the feedback system by implementing only one set of eigenmode matrices for all values of signal-to-noise ratio (SNR) and a much smaller set of power division matrices that differ slightly for different values of SNR. As a result, the required feedback bit rate is kept low and constant throughout the whole range of SNR values of interest. The additional advantage of the splitting of the modulation matrix into two parts is that it can lower the feedback bit rate for slowly-varying channels. If the eigenmodes of the channel stay within the same region for an extended period of time, additionally, nested encoding can be performed to increase the resolution of the CSTT and improve the system capacity.

The actual design of the receiver and transmitter eigenmode and power division matrices can be done using numerical or analytical methods and is not the object of this disclosure. However, our method allows for actual implementations of systems closely approaching the theoretical capacities of MIMO channels without putting any unrealistic demand on the feedback link throughput. This is a major improvement compared to the other state-of-the art CSI quantization methods, which experience problems approaching the theoretical capacities and suffer from the carly onset of capacity ceiling due to inter-user interference at relatively low SNR.

I. System Model for Single User Communication and OFDM

We assume that the communication system consists of a transmitter equipped with n_(T) antennas[22] and a receiver[24] with n_(R) antennas[26]. A general frequency selective fading channel is modeled by a set of channel matrices H_(j) of dimension n_(R)×n_(T) defined for each sub-carrier j=0,1 . . . N_(OFDM)-1. The received signal at the jth sub-carrier is then given by the n_(R)-dimensional vector y_(j) defined as

y _(j) =H _(j) x _(j) +n _(j)   (1)

where x, is the n_(T)-dimensional vector of the transmitted signal and n_(j) is the n_(R)-dimensional vector consisting of independent circular complex Gaussian entries with zero mean and variance 1. Moreover, we assume that power is allocated equally across all sub-carriers |x_(j)|²=P.

II. Quantizing Water-Filling Information

If the transmitter has access to the perfect channel state information about the matrix H_(j), it can select the signaling vector x_(j) to maximize the closed-loop system capacity

$\begin{matrix} {C = {\sum\limits_{j}{\log_{2}{\det\left\lbrack {I + {H_{j}Q_{j}H_{j}^{H}}} \right\rbrack}}}} & (2) \end{matrix}$

where Q_(j)=E[x_(j)x_(j) ^(H)]. Unfortunately, optimizing the capacity in (2) requires a very large feedback rate to transmit information about optimum Q_(j) (or correspondingly H_(j)) which is impractical. Instead, we propose using a limited feedback link, with the transmitter choosing from a set of matrices {circumflex over (Q)}(n).

Using the typical approach involving singular value decomposition and optimum water-filling, we can rewrite (1) as

y _(j) =H _(j) x _(j) +n _(j)=(U _(j) D _(j) V _(j) ^(H))(V _(j) {tilde over (x)} _(j))+n _(j)   (3)

where E[{tilde over (x)}_(j){tilde over (x)}_(j) ^(H)]; constrained with Tr (S_(j))=P is a diagonal matrix describing optimum power allocation between the eigenmodes in V_(j). Based on (3), the set of matrices Q_(j)=V_(j)S_(j)V_(j) ^(H), maximizes capacity in (2).

To construct the most efficient vector quantizer for channel feedback, the straightforward approach would be to jointly optimize signal covariance matrices {circumflex over (Q)} for all sub-carriers at once. Such an approach, however, is both complex and impractical, since any change of channel description and/or power level would render the optimized quantizer suboptimal. Instead, we propose an algorithm which separately quantizes information about eigenmode matrices V_(j) in codebook {circumflex over (V)} and power allocation S_(j) in codebook Ŝ. Note that the first variable depends only on channel description and not on the power level P which simplifies the design.

We optimize the quantizers {circumflex over (V)} and Ŝ for flat-fading case and we apply them separately for each sub-carrier in case of OFDM modulation. Although such an approach is sub-optimal, it allows a large degree of flexibility since different system setups can be supported with the same basic structure.

A. Quantizing Eigenmodes

We assume that the receiver[24] has perfect channel state information (CSIR) and attempts to separate[30] the eigenmode streams {tilde over (x)}_(j) in (3) by multiplying y_(j) with U_(j) ^(H). However, if the transmitter uses quantized eigenmode matrix set with limited cardinality, the diagonalization of {tilde over (x)}_(j) will not be perfect. To model this, we introduce a heuristic distortion metric which is expressed as

γv(n; H)=∥DV ^(H) {circumflex over (V)}(n)−D∥ _(F)   (4)

where {circumflex over (V)}(n) is the nth entry in the predefined set of channel diagonalization matrices and ∥·∥_(F) is the Frobenius norm. We omitted subscript entries j in (4) for the clarity of presentation.

We assume that n=0,1 . . . 2^(N) ^(v) −1 where N_(v) is the number of bits per channel realization in the feedback link needed to represent the vectors {circumflex over (V)}(n). To design the quantizer using (4), we divide the whole space of channel realizations H into 2^(N) ^(v) regions V_(i) where

V_(i)={H:γv(i;H)<γv(j;H) for all j≠i}.   (5)

It can be shown that minimizing this metric should, on average, lead to maximizing the ergodic capacity of the channel with the quantized feedback (when γ(n;H)=0 the channel becomes perfectly diagonalized). The optimum selection of {circumflex over (V)} and regions V_(i) in (5) is an object of our current work. Here, however, we use a simple iterative heuristic based on a modified form of the Lloyd algorithm, which has very good convergence properties and usually yields good results. The algorithm starts by creating a codebook of centroids {circumflex over (V)} and, based on these results, divides the quantization space into regions V_(i). The codebook is created as follows:[50]

-   -   1. Create a large training set of I random matrices H(l).[46]     -   2. For each random matrix H(l), perform singular value         decomposition to obtain D(l) and V(l) as in (3).     -   3. Set iteration counter i=0. Create a set of 2^(N) ^(v) random         matrices Ĥ(n).     -   4. For each matrix Ĥ(n) calculate corresponding {circumflex over         (V)}^((i)) (n) using singular value decomposition.     -   5. For each training element H(I) and codebook entry {circumflex         over (V)}^((i)) (n) calculate the metric in (4). For every l         choose indexes n_(opt)(l) corresponding to the lowest values of         γv(n;H(l)).     -   6. Calculate a new set {circumflex over (V)}^((i+1)) (n) as a         form of spherical average of all entries V(l) corresponding to         the same index n using the following method. (The direct         averaging is impossible since it does not preserve orthogonality         between eigenvectors.) For all a calculate the subsets L(n)=(l :         n_(opt)(l)=n) and if their respective cardinalities |L(n)|≠0 the         corresponding matrices Q ^((i+1))(n) can be obtained as

$\begin{matrix} {{{\overset{\_}{Q}}^{({i + 1})}(n)} = {\frac{1}{❘{L(n)}❘}{\sum\limits_{l \in {L(n)}}{{V(l)}^{1}{{OV}(l)}^{H}}}}} & (6) \end{matrix}$

where ^(I)O is an n_(T)×n_(T) all-zero matrix with the exception of the upper-left corner element equal to 1. Finally, using singular value decomposition, calculate {circumflex over (V)}^((i+1))(n) from

Q ^((i+1))={circumflex over (V)}^((i+1))(n)W({circumflex over (V)}^((i+1))(n))^(H)   (7)

where W is a dummy variable.

-   -   7. Calculate the average distortion metric

γ _(v) ^((i+1))=1/L Σ_(l)γv(n_(opt)(l); H(l)).

-   -   8. If distortion metric fulfills |γ _(v) ^((i+1))−γ _(v)         ^((i))|/γ _(v) ^((i))<Θ, stop. Otherwise increase i by 1 and go         to 5).

Upon completion of the above algorithm, the set of vectors {circumflex over (V)} can be used to calculate the regions in (5). The results of the codebook optimization are presented in FIG. 11A for a case of ny=n =2 and Ny=2. The optimization was performed using L=1,000·2^(N) ^(v) and ⊖=10⁻⁷.

B. Quantizing Power Allocation Vectors

Having optimized[50] power-independent entries in the codebook of channel cigenmode matrices {circumflex over (V)}, the next step is to create a codebook for power allocation Ŝ[118]. We use a distortion metric defined as

$\begin{matrix} {{\gamma{s\left( {k;H;P} \right)}} = \frac{\det\left\lbrack {I + {HQH}^{H}} \right\rbrack}{\det\left\lbrack {I + {H{\hat{V}\left( n_{opt} \right)}{\hat{S}(k)}{{\hat{V}}^{H}\left( n_{opt} \right)}H^{H}}} \right\rbrack}} & (8) \end{matrix}$

where Ŝ(k) is the kth entry in the predefined set of channel water-filling matrices and V (n_(opt)) is the entry in the {circumflex over (V)} codebook that minimizes metric (4) for the given H. We use k =0,1, . . . 2^(N) ^(s) −1 where N_(S) is the number of bits per channel realization in the feedback link needed to represent the vectors Ŝ(k). Minimizing the metric in (8) is equivalent to minimizing the capacity loss between the optimum water-filling using Q and the quantized water-filling using {circumflex over (V)} and Ŝ.

Similarly to the previous problem, we divide the whole space of channel realizations H into 2^(N) ^(s) regions S_(i)(P) where

S _(i)(P)={H: γs(i;H;P)<γs(j;H;P) for all j≠i}.   (9)

and to create the codebook Ŝ, we use the following method:

-   -   1. Create a large training set of L random matrices H(l).     -   2. For each random matrix H(l), perform water-filling operation         to obtain optimum covariance matrices Q(l) and S(l).     -   3. Set iteration counter i=0. Create [100],[104] a set of 2^(N)         ^(s) random diagonal matrices Ŝ^((i))(k) with Tr(Ŝ^((i))(k))=P.     -   4. For every codebook entry Ŝ^((i))(k) and matrix Q(l)         calculate[112] the metric as in (8). Choose[106] indexes         k_(opt)(l) corresponding to the lowest values of         γ_(S)(k)H(l);P).     -   5. If γ_(S)(k_(opt)(l);(H(l);P)>γ_(eq) ^((H(l);P)) where γ_(eq)         ^((H(l);P)) is the metric corresponding to equal-power         distribution defined as

$\begin{matrix} {{{\gamma{{eq}\left( {{H(l)};P} \right)}} = \frac{\det\left\lbrack {I + {{H(l)}{Q(l)}{H^{H}(l)}}} \right\rbrack}{\det\left\lbrack {I + {P/n_{T}{H(l)}{H^{H}(l)}}} \right\rbrack}},} & (10) \end{matrix}$

set the corresponding entry kope(!) =2^(N) ^(s) . For all k calculate the subsets[108]L(k)={l: k_(opt)(l)=k}.

-   -   6. For all k=0,1, . . . 2^(N) ^(s) −1[114] for which |L(k)|≠0,         calculate[16] a new Ŝ^((i+1))(k) as the arithmetic average

$\begin{matrix} {{{\hat{S}}^{({i + 1})}(k)} = {\frac{1}{❘{L(k)}❘}{\sum\limits_{l \in {L(k)}}{S(l)}}}} & (11) \end{matrix}$

-   -   7. Calculate the average distortion metric

$\begin{matrix} {{\overset{\_}{\gamma}}_{s}^{({i + 1})} = {\frac{1}{L}{\sum\limits_{l}{\min\left\{ {{\gamma{s\left( {{k_{opt}(l)};{{H(l)}P}} \right)}},{\gamma_{eq}\left( {{H(l)}P} \right)}} \right\}}}}} & (12) \end{matrix}$

-   -   8. If distortion metric fulfills |γ _(v) ^((i+1))−γ _(v)         ^((i))|/γ _(v) ^((i))<Θ stop. Otherwise increase i with 1 and go         to 4).

The set of vectors Ŝ is then used to calculate the regions in (9). Since waterfilling strongly depends on the power level P and {circumflex over (V)}, optimally the Ŝ should be created for every power level and number of bits N_(v) in eigenvector matrix codebook. As an example, the results of the above optimization are presented in FIG. 11B for a case of n_(T)=n_(R)=2, N_(V)=2 and N_(S)=1. The optimization was performed using L=1,000·2^(N) ^(s) and ⊖=10⁷. The curves show the entries on the diagonals of the two matrices Ŝ(k) normalized with P. As one can see, one of the matrices tends to assign all the power to one of the eigenmodes, while the other balances the assignment between them. As expected, the balancing becomes more even with increasing P where the capacity of the equal-power open-loop systems approaches the capacity of the water-filling closed-loop systems. Since the differences between the entries of Ŝ(k) are not that large for varying powers, it is possible to create an average codebook Ŝ which could be used for all values of P but we do not treat this problem in here.

An interesting property of the above algorithm is that it automatically adjusts the number of entries in Ŝ according to the number of entries in {circumflex over (V)}. For low values of Ny, even if the algorithm for selection of Ŝ is started with high N_(S), the optimization process will reduce the search space by reducing cardinality |L(k)| of certain entries to 0. As & result, for N_(V)=2,3, N_(S)=1 will suffice, while for N_(V)=4, the algorithm will usually converge to N_(S)=2. This behavior can be easily explained since for low resolution of the channel eigenvector maurices {circumflex over (V)} only low precision is necessary for describing Ŝ. Only with increasing N_(V), the precision N_(S) becomes useful.

III. Vq Algorithm For Flat-Fading Mimo Channels

The vector quantizers from the previous sections are first applied to a flat-fading channel case. In such a case, the elements of each matrix H in (1) are independent circular complex Gaussian elements, normalized to unit variance.

The system operation can now be described as follows:

-   -   1. The receiver[24] estimates the channel matrix H.     -   2. The receiver[24] localizes the region V_(i) according to (5)         and stores its index as n_(opt)[32]     -   3. Using n_(opt), the receiver[24] places H in a region S_(i)         according to (9) and stores its index as k_(opt).     -   4. If the resulting system capacity using the predefined         codebook entries is higher than the capacity of equal power         distribution as in

C(n _(opt) , k _(opt))>log₂det[I+P/n _(T) HH ^(H)]  (13)

indexes n_(opt) and k_(opt) are fed back to the transmitter. [34],[36]

-   -   5. The transmitter uses[40],[38A] the received indices of a         codebook entries to process its signal. If there is no feedback,         power is distributed equally between the antennas[22].

Using the above algorithm, the system's performance is lower-bounded by the performance of the corresponding open-loop system and improves if the receiver finds a good match between the channel realization and the existing codebook entries. The salient advantage of such an approach is its flexibility and robustness to the changes of channel model. If there are no good matches in the codebook, the feedback link is not wasted and the transmitter uses the equal power distribution. The disadvantage of the system is that the feedback link is characterized by a variable bit rate.

IV. Vq Algorithm For Frequency-Selective MIMO-OFDM Channels

In case of the frequency-selective channel, flat fading algorithm is applied to the separate OFDM sub-carriers. Although this approach is clearly sub-optimal, it allows us to use a generic vector quantizer trained to the typical flat-fading channel in a variety of other channels.

In general case, the feedback rate for such an approach would be upperbounded by N_(OFDM)(N_(V)+N_(D)). However, as pointed out by Kim et al., the correlation between the adjacent sub-carriers in OFDM systems can be exploited to reduce the required feedback bit rate by proper interpolating between the corresponding optimum signalling vectors.

In this work, we use a simpler method which allows the receiver[24] to simply group adjacent M sub-carriers and perform joint optimization using the same codebook entry for all of them (such methods are sometimes called clustering).

V. Simulation Results

A. Flat-fading channel

We tested the system on 2×2 MIMO and 4×4 MIMO channels with varying SNR and feedback rates. We tested 2×2 MIMO channel with N_(V)=2,3,4 and N_(D)=1, corresponding to total feedback rate of between 3 and 5 bits. Correspondingly, in case of 4×4 MIMO, we used N_(V)=10,12,14 and N_(D)=2, corresponding to total feedback rate between 12 and 16 bits. We define an additional parameter called feedback frequency, v which defines how often the receiver[24] requests a specific codebook entry instead of equal power distribution and an average feedback bit rate as R_(b)=v(N_(V)=N_(S)).

FIG. 12(a) presents the results of simulations of ergodic capacity of the system (based on 100,000 independent channel matrices H) in case of perfect CSIT, vector quantized feedback (partial CSIT) and no CSIT. It is clearly seen that, even for very low bit rates on a feedback channel, the proposed scheme performs closely to the optimum. A rule of thumb seems to be that the number of bits needed to encode the codebook is approximately equal to n_(T)×n_(R). Moreover, FIG. 12(b) shows that as the SNR grows, less feedback is required to provide good system performance and the proposed algorithm automatically reduces the reverse link usage.

It is also interesting to note that increasing the quality of quantization increases the feedback frequency v. This is a consequence of the fact that there is a higher probability of finding a good transmit signal covariance matrix when there are a lot of entries in the codebook.

B. Frequency-Selective Channel

We have simulated the 2×2 MIMO system using the OFDM modulation with carrier frequency: f_(c)=2 GHz; signal bandwidth; B=5 MHZ, number of sub-carriers: N_(OFDM)=256; ITU-R M.1225 vehicular A channel model with independent channels for all pairs of transmit and receive antennas[22][26]; the guard interval equal to the maximum channel delay.

The results of simulations are presented in FIGS. 13A and 13B. Since the capacity curves of this system are very similar to capacity of the flat-fading 2×2 system in FIGS. 12A and 12B we decided to show the losses of performance as compared to the perfect water-filling case instead. FIG. 13A shows the loss of performance defined as C-C(M) where C defined in (2) and C(N_(V), N_(S), M) is the capacity of the system with different feedback rates and clustering of M sub-carriers. As expected, increasing the clustering, decreases the throughput since the same covariance matrix is used for too many adjacent sub-carriers. At the same time, in FIG. 13B shows that the required average feedback rate decreases significantly with increasing M. For the simulated channel, the best results are provided by M=8, which is approximately equal to the coherence bandwidth of the channel. An interesting feature of the OFDM-MIMO is that, unlike in the flat-fading case, the feedback rate remains almost constant throughput the P range. In any case, however, around two orders of magnitude more feedback bit rate is required on frequency selective channel.

VI. System Model For Multi User Communication

We assume that the communication system consists of a transmitter equipped with n_(T) antennas[22] and K≥n_(T) mobile receivers[24] with identical statistical properties and n_(R)(k) antennas[26], where k=1,2 . . . K. The mobile user channels are modeled by a set of i.i.d. complex Gaussian channel matrices H_(k) of dimension n_(R)(k)×n_(T). (Throughout the document we use the upper-case bold letters to denote matrices and lower-case bold letters to denote vectors.) The received signal of the kth user is then given by the n_(R)(k)-dimensional vector y_(k) defined as

y _(k) =H _(k) x+n _(k)   (14)

where x is the n_(T)-dimensional vector of the transmitted signal and n_(k) is the n_(R)(k)-dimensional vector consisting of independent circular complex Gaussian entries with zero means and unit variances. Finally, we assume that the total transmit power at each transmission instant is equal to P. The above assumptions cover a wide class of wireless systems and can easily be further expanded to include orthogonal frequency division multiplexing (OFDM) on frequency-selective channels or users with different received powers (due to varying path loss and shadowing).

Although theoretically it is possible to design the optimum CSI quantizer for the above canonical version of the system, such an approach may be impractical. For example, subsets of receivers[24] with different numbers of receive antennas[26] would require different CSI codebooks and quantizer design for such a system would be very complex. To alleviate this problem, we assume that the base station treats each user as if it was equipped with only one antenna[26], regardless of the actual number of antennas[26] it may have. While suboptimal, such an approach allows any type of a receiver[24] to work with any base station and may be even used to reduce the quantization noise as shown by Jindal. We call such system setup virtual multiple-input single-output (MISO) since, even though physically each transmitter-receiver link may be a MIMO link, from the base station's perspective it behaves like MISO.

We follow the approach of Spencer et al., where each user performs singular value decomposition of H_(k)=U_(k)S_(k)V_(k) ^(H) [30] and converts its respective H_(k) to a n_(T)-dimensional vector h_(k) as

b _(k) =u _(k) ^(H) J _(k) =s _(k) ^(max) v _(k) ^(h)   (15)

where s_(k) ^(max) is the largest singular value[30A] of S_(k) and u_(k) and v_(k) are its corresponding vectors[30] from the unitary matrices U_(k) and V_(k), respectively.

Based on (15), the only information that is fed[36],[36A] back from[34],[34A] the receivers[24] to the transmitter is the information about the vectors h_(k), which vastly simplifies the system design and allows for easy extensions. For example, if multiple streanis per receiver are allowed, the channel information for each stream can be quantized using exactly the same algorithm.

VII. System Design With Full Csit

In this section, we present typical approaches for the system design when full CSIT is available. As a simple form of multi-user selection diversity, we define a subset of active users with cardinality n_(T) as S. Furthermore, for each subset S, we define a matrix H(S)=[h₁ ^(T), h₂ ^(T), . . . , h_(n) _(T) ^(T), ]^(T), whose rows are equal to the channel vectors h_(k) of the active users.

A. Cooperative Receivers

The upper -bound for system sum-rate is obtained when the users are assumed to be able to cooperate. With such an assumption, it is possible to perform singular value decomposition of the joint channel as H[S]=U[S]S[S]V^(H)[S]. Defining s; as the entries 5 on the diagonal of S[S] allows to calculate the maximum sum rate of a cooperative system as

$\begin{matrix} {R^{coop} = {\max\limits_{S}{\sum\limits_{i = 1}^{n_{T}}\left\lbrack {\log_{2}\left( {{\xi\lbrack S\rbrack}s_{i}^{2}} \right)} \right\rbrack_{+}}}} & (16) \end{matrix}$

where ξ[S] is the solution of the water-filling equation Σ_(i=1) ^(n) ^(T) [ξ[S]−1/s_(l) ²]₊=P.

B. Zero-Forcing Dirty-Paper Coding

In practice, the receivers[24] cannot cooperate and the full diagonalization of the matrix H[S] is impossible. The problem can still be solved by using linear zero-forcing (ZF) followed by non-linear dirty paper precoding, which effectively diagonalizes the channels to the active users. The matrix H[S] is first QR-decomposed as H[S]=L[S]Q[S], where L[S] is lower triangular matrix and Q[S] is a unitary matrix. After multiplying the input vector x by Q[S]^(H), the resulting channel is equal to L[S], i.e., the first user does not suffer from any multi-user interference (MUI), the second user receives interference only from the first user, etc.

In this case, non-causal knowledge of the previously encoded signals can be used in DPC encoder allowing the signal for each receiver[24] i>1 to be constructed in such a way that the previously encoded signals for users k<i, are effectively canceled at the ith receiver[24]. Since the effective channel matrix is lower triangular, the channel will be diagonalized after the DPC, with l_(i) being the entries on the diagonal of LAS). This leads to maximum sum-rate calculation as

$\begin{matrix} {R^{{zf} - {dpc}} = {\max\limits_{S^{ord}}{\sum\limits_{i = 1}^{n_{T}}\left\lbrack {\log_{2}\left( {{\xi\left\lbrack S^{ord} \right\rbrack}l_{i}^{2}} \right)} \right\rbrack_{+}}}} & (17) \end{matrix}$

where ξ[S^(ord)] is the solution of the water-filling equation. Note that, as opposed to (16), the maximization is performed over ordered versions of the active sets S.

C. Linear Modulation

Even though, theoretically, the above approach solves the problem of the receiver[24] non-cooperation, its inherent problem is the absence of effective, low complexity DPC algorithms. Moreover, since dirty-paper coding requires full CSIT it is likely that systems employing DPC would require significantly higher quality of channel feedback than simpler, linear precoding systems.

We use the linear block diagonalization approach, which eliminates MUI by composing the modulation matrix B[S] of properly chosen null-space eigenmodes for each set S. For each receiver[24] i∈S, the ith row of the matrix H[S] is first deleted to form H[S_(i)]. In the next step, the singular value decomposition is performed[30],[30A] to yield H[S_(i)]=U[S_(i)]S[S_(i)]V^(H)[S_(i)]. By setting the ith column of B[S] to be equal to the

rightmost vector of V[S_(i)], we force the signal to the ith receiver[24] to be transmitted in the null-space of the other users and no MUI will appear. In other words, the channel will be diagonalized with di being the entries on the diagonal of H[S]B[S]. This leads to formula

$\begin{matrix} {R^{linear} = {\max\limits_{S}{\sum\limits_{i = 1}^{n_{T}}\left\lbrack {\log_{2}\left( {{\xi\lbrack S\rbrack}d_{i}^{2}} \right)} \right\rbrack_{+}}}} & (18) \end{matrix}$

where ξ(S) is the solution of the water-filling equation.

As an example, FIG. 14 shows the sum-rates of the discussed systems for K=10 users and different antenna configurations. The zero-forcing DPC system approaches the cooperative system's sum-rate as the transmitted power increases. The sub-optimal linear modulation provides lower sum-rate but losses at P>0 dB, as compared to the ZF-DPC system, are in the range of only 1-2 dB for the 4×4 configuration and fractions of dB for the 2×2 system. Since the linear system is much easier to implement than ZF-DPC, we will use it to test our CSI encoding algorithms.

VIII. System Design With Partial Csit

The systems discussed so far are usually analyzed with assumption that, at any given time, the transmitter will have full information about the matrices H[S]. Unfortunately, such an assumption is rather unrealistic and imperfect CSIT may render solotions relying on full CSIT useless.

In practice, the receivers[24] will quantize the information about their effective channel vectors h_(k)[30] as ĥ_(k)[32], according to some optimization criterion. Based on this information, the transmitter will select[38],[52],[62] the best available modulation matrix {circumflex over (B)} from the predefined transmitter codebook and perform water-filling using the best predefined power division matrix {circumflex over (D)}. Regardless of the optimization criterion, the finite cardinality of the vector codebooks will increase MUI and lower system throughput. FIG. 15 shows the mechanism leading to appearance of the MUI in a simple system with n_(T)=2 and K=2 users with effective channel vectors h₁ and h₂, encoded by the quantizer as ĥ₁ and ĥ₂. If the full CSIT is available, the transmitter will choose[38] a modulation matrix based on eigenmodes v₁ and v₂, which are perpendicular to vectors h₂ and hy, respectively. As a result, each user will be able to extract its desired signal without MUI. With partial CSIT, however, the transmitter will obtain only approximate versions of the effective channel vectors, and the resulting modulation matrix will be based on eigenmodes {circumflex over (v)}₁ and {circumflex over (v)}₂. whose dot products with h₂ and h₁ will not be zero, causing the MUI.

IX. CSI Quantization Algorithms

The fundamental difference between CSI encoding in single-user and multiple-user systems is that during normal system operation, each receiver[24] chooses its vector ĥ_(k) without any cooperation with other receivers[24]. This means that the design of optimum codebook for h_(k) must precede the design of codebooks {circumflex over (B)} and {circumflex over (D)}. Based on (15), one can see that channel state information in form of the vector hy consists of the scalar value of channel gain[30A] s_(k) ^(max) and the cigenmode[30] v_(k) ^(H). Since these values are independent, we propose an algorithm which separately quantizes the information about cigenmodes[32] in codebook {circumflex over (v)} and amplitude gains[32A] in codebook ŝ.

A. Quantization[32],[5] of Receiver Channel Eigenmodes

We assume that N_(v) is the number of bits per channel realization in the feedback link needed to represent the vectors v_(k) in (15). We divide the space of all possible v's into 2^(N) ^(v) regions v_(i)

v_(i)={v;γ_(v)(i;v)<γ_(v)(j;v) for all j≠i}  (19)

where γ_(v)(n; v) is a distortion function. Within each region va we define a centroid vector {circumflex over (v)}(i)[49], which will be used as a representation of the region. The design of the codebook {circumflex over (v)} can be done analytically and/or heuristically using for example the Lloyd algorithm. In this work, we define the distortion fonction as the angle between the actual vector v and {circumflex over (v)}(i): γ_(v)(i;v) 32 cos⁻¹({circumflex over (v)}(i)·v), which has been shown by Roh and Rhao to

maximize ergodie capacity, and use Lloyd algorithm to train[47] the vector quantizer. Note that the construction of {circumflex over (v)} is independent of the transmit power.

B. Quantization[32A] Of Receiver Amplitude Gains

We assume that N_(s) is the number of bits per channel realization in the feedback link needed to represent the scalar s_(k) ^(max) in (15). We divide the space of all possible channel realizations s=s_(k) ^(max) into 2As regions s_(i)

  (20)

where ŝ(i) are scalar centroids representing regions s_(i). In this work, we perform the design[102] of the codebook ŝ using the classical non-uniform quantizer design algorithm with distortion function given by quadratic function of the quantization error as ϵ(i;s)=(s−ŝ(i))².

The construction of the codebook s is generally dependent on the transmit power level. However, as pointed out above the differences between the codebooks ŝ for different power regions are quite small. This allows us to create only one codebook ŝ and use it for all transmit powers.

C. Quantization of the Transmitter Modulation Matrices

The calculation of the modulation matrix {circumflex over (B)} is based on the given codebook {circumflex over (v)}. We assume that the quantization[32] of the channel eigenmodes is performed at the receiver[24] side and each user transmits[34] back its codebook index if. The indices are then used at the transmitter side to select[38][52][62] the modulation matrix {circumflex over (B)}(i₁, i₂. . . i_(k)). Since, from the linear transmitter point of view, ordering of the users is not important, we will use the convention that the indices (i₁, i₂. . . i_(k)) are always presented in the ascending order. For example, in a system with K=2, n_(T)=2 and I-bit vector quantizers {circumflex over (v)}, there will exist only three possible: modulation matrices corresponding to sets of {circumflex over (v)} indices (1,1), (1,2) and (2,2).

In the context of vector quantizing, the design of the modulation matrices can no longer be based on the algorithm presented in Section VII.C. Using this method with quantized versions of h_(k) produces wrong result when identical indices i_(k), are returned and the receiver[24] attempts to jointly optimize transmission to the users with seemingly identical channel vectors h_(k). Instead, we propose the following algorithm to optimize the set of matrices {circumflex over (B)}(i₁, i₂. . . i_(K)):

-   -   1. Create a large set of Nn_(T) random matrices[46] H_(k), where         N is the number of training sets with n_(T) users each.     -   2. For each random matrix H_(k), perform singular value         decomposition[68] and obtain h([70] as in (15).     -   3. For each vector h_(k) store[74] the index i_(k) of the         corresponding entry {circumflex over (v)}(i_(k l ).)     -   4. Divide[76] the entire set of matrices H_(k) into N sets with         n_(T) elements each.     -   5. Son[78] the indices is within each set l in the ascending         order. Map[78] all unique sets of sorted indices to a set of         unique indices I_(B)(for example (1,1)→l_(B)=1; (1,2)→l_(B)=2;         (2,2)→l_(B)=3 . . . ).     -   6. In each set l, reorder the corresponding channel vectors he         according to their indices i_(k) and calculate[80] the optimom         B_(l) using the method from Section VII.C.     -   7. Calculate[84] a set B (Ip) as a column-wise spherical average         of all entries B_(l) corresponding to the same[82] index l_(B).

After calculation of |l_(B)| modulation matrices {circumflex over (B)}, the remaining part of system design is the calculation of the water filling matrices {circumflex over (D)}, which divide the powers between the eigenmodes at the transmitter. The procedure for creation of codebook {circumflex over (D)}[118] is similar to the above algorithm, with the difference that the entries ŝ(n_(k)) are used instead of {circumflex over (v)}(i_(k)), and the spherical averaging of the water-filling matrices is performed diagonally, not column-wise. Explicitly:

-   -   1. Create a large set of Nn_(T) random matrices[46] H_(k), where         N is the number of training sets with n_(T) users each.     -   2. For each random matrix H_(k), perform singular value         decomposition[ 104] and obtain h_(k) as in (15).     -   3. For each vector hy store the index n_(k) of the corresponding         entry ŝ(n_(k))[1106]     -   4. Divide the entire set of matrices H_(k) into N sets with ny         elements cach.     -   5. Sort the indices n_(k) within each set l in the ascending         order. Map all unique sets of sorted indices to a set of unique         indices Ip (for example (1,1)→l_(D)=1; (1,2)→l_(D)=2;         (2,2)→l_(D)=3 . . . ).[110]     -   6. In each set l, reorder the corresponding channel vectors         h_(k) according to their indices n_(k) and calculate the optimum         D_(l) using the method of water-filling from Section VII.C.[112]     -   7. Calculate[116] a set ((In) as a diagonal spherical average of         all entries D; corresponding to the same[114] index I_(D).

D. System Operation

The matrices {circumflex over (B)} and {circumflex over (D)} are used in the actual system in the following way:

-   -   1. The K mobile receivers[24] estimate[30],[30A] their channels         and send the indices i_(k)[34] and n_(k)[34A] of the         corresponding receiver quantizer entries {circumflex over         (v)}(i_(k) )[32] and ŝ(n_(k) )[32A] to the base station.     -   2. The transmitter forms I sets of users corresponding to all         combinations of n_(T) users out of K. Within each set l, the         indices i_(k)[58] and n_(k)[63] are sorted in the ascending         order and mapped to their respective indices l_(B)(l)[60] and         l_(D)(l)[64];     -   3. Within each set l, the matrices {circumflex over         (B)}[l_(B)(l)}[52],[62] and {circumflex over (D)}(l_(D)(l)]         [54],[38A] are used to estimate[56],[66] instantaneous sumo-rate         R(l).     -   4. The base station flags the set of users providing highest         R(l) as active for the next transmission epoch.     -   5. The transmitter uses the selected matrices to transmit         information.

E. Nested Quantization of Channel Eigenmodes

The above algorithm does not assume any previous knowledge of the channel and the feedback rate required to initially acquire the channel may be high. In order to reduce it on slowly varying channels, we propose a nested quantization method shown in FIGS. 16A and 16B. In this example, an initial coarse quantization of the CSI is performed[88] using only 2 bits. Assuming[98] that the actual channel vector lies in region v₃ and that it stays within this region during the transmission of subsequent frames[90],[92],[94], it is possible to further quantize v₃ using nested, precise quantization[96]. In this way, the effective feedback rate is still 2 bits, but the resolution corresponds to a 4-bit quantizer. In order to quantify the possibility of such a solution, we introduce eigenmode coherence time τ_(eig)(N_(v)), which, related to the frame duration T_(frame), shows for how long the channel realization stays within the same region v_(i) of the N_(v)-bit quantizer. Notice that eigenmode 10) coherence time depends on the number of bits N_(v): the higher the initial VQ resolution, the faster the channel vector will move to another region and the benefits of nested quantization will vanish.

X. Simulation Results

We have implemented our system using a base station with n_(T)=2 and a set of K mobile receivers[24] with identical statistical properties and n_(R)(k)=n_(T)=2. We have varied the number of users from 2 to 10 and optimized vector quantizers using methods presented above, Each system setup has been simulated using 10,000 independent channel realizations.

FIGS. 17 and 18 show the results of the simulations for varying feedback rates. Except for very high transmit power values P>15 dB, it is possible to closely approach the sum-rate of the full CSI system with 8 bits (N_(v)=7, n_(s)=1) in the feedback link. The required feedback rate N_(v) is much higher than N_(s), which shows the importance of high quality eigenmode representation in multiuser systems. In high power region, increasing N_(v) by 1 bit increases the spectral efficiency by approximately 1 bit/channel use.

FIG. 19 shows the results of nested quantization with low feedback rates when the channel's eigenmode coherence time is longer than frame duration. If such a situation occurs, the considered system may approach the theoretical full CSIT sum-rate using only 5 bits per channel use in the feedback link.

Note that further feedback rate reduction can be achieved with the algorithm presented by Jindal. However, we will not present these results here.

XI. Additional Matter

In case of multiple user systems, multi-user diversity may be achieved by a simple time-division multiplexing mode (when only one user at a time is given the full bandwidth of the channel) or scheduling the transmission to multiple users[24] at a time. Here we analyze the former approach and assume that the base station will schedule only one user[24] for transmission.

If the system throughput maximization is the main objective of the system design, the transmitter must be able to estimate[56] the throughput of each of the users, given the codebook indices it received from each of them. Assuming that the kth user returned indices requesting the eigenmode codeword {circumflex over (V)}_(k) and power allocation codeword Ŝ_(k), the user's actual throughput is given as

R _(k) ^(single)=log₂det[I _(nR(k)) +H _(k){circumflex over (B)}_(k)Ŝ_(k){circumflex over (V)}_(k) ^(H) H _(k) ^(H)]  (21)

Using singular value decomposition of channel matrix H_(k) and equality det[I_(NR(k))+H_(k){circumflex over (Q)}_(k)H_(k) ^(H)]=det[I_(nT)+{circumflex over (Q)}_(k)H_(k) ^(H)H_(k)], it can be shown that

R _(k) ^(single)=log₂det[I _(nT(k)) +H _(k){circumflex over (B)}_(k)Ŝ_(k){circumflex over (V)}_(k) ^(H) H _(k) ^(H)]=log₂det[I_(nR(k))+Ŝ_(k) E _(k) ^(H) D _(k) ² E _(k)]  (22)

where E_(k)=V_(k) ^(H){circumflex over (V)}_(k)is a matrix representing the match between the actual eigenmode matrix of the channel and its quantized representation (with perfect match E_(k)=I_(NT)).

In practice, the actual realization of E_(k) will not be known at the transmitter, and its mean quantized value Ê_(k), matched to {circumflex over (V)}_(k) must be used instead. Similarly, the transmitter must use a quantized mean value {circumflex over (D)}_(k)which is matched to the reported water-filling matrix Ŝ_(k). This leads to the selection criterion for the optimum user[24] k_(pt)

$\begin{matrix} {k_{opt} = {\arg\max\limits_{{k = 1},2,{\ldots K}}\log_{2}{{\det\left\lbrack {I_{n_{R}(k)} + {{\hat{S}}_{k}{\hat{E}}_{k}^{H}{\hat{D}}_{k}^{2}{\hat{E}}_{k}}} \right\rbrack}.}}} & (23) \end{matrix}$

Similarly to single user selection, also in the case of multi-user selection the choice of active users must be made based on incomplete CSIT. The quantized CSI will result in appearance of multi-user interference. We represent this situation using variable which models the dot product of the quantized eigenmode {circumflex over (v)}_(k) ^(n) reported by the kth user in the set S, and the lth vector in the selected modulation matrix[52] {circumflex over (B)}_(S).

Moreover, assuming that the quantized singular value of the kth user in the set S is given by {circumflex over (d)}_(k) and the transmitter uses power allocation matrix[54] Ŝ_(S), the estimated sum-rate of the subset S is given as[56]

$\begin{matrix} {{R^{multi}(S)} = {\sum\limits_{k \in S}{{\log_{2}\left( {1 + \frac{P{{\hat{d}}_{k}^{2}\left\lbrack {\hat{S}}_{S} \right\rbrack}_{k,k}{\hat{E}}_{k,k}^{2}}{1 + {P{\hat{d}}_{k}^{2}{{\sum}_{l \neq k}\left\lbrack {\hat{S}}_{S} \right\rbrack}_{l,l}{\hat{E}}_{k,l}^{2}}}} \right)}.}}} & (24) \end{matrix}$

Note that, due to the finite resolution of the vector quantizer, the multi-user interference will lower the max sum-rate R^(multi)(S)<R^(max) for all S.

Based on (24) the choice of the active set of users is then performed as

$\begin{matrix} {S_{opt} = {\arg\limits_{S}\max{{R^{vq}(S)}.}}} & (25) \end{matrix}$

One can also modify the algorithm presented, in section II.A as follows: we use a simple iterative heuristic based on a modified form of the Lloyd algorithm, which has very good convergence properties. The algorithm starts by creating a random codebook of centroids {circumflex over (V)} and iteratively updates it until the mean distortion metric changes become smaller than a given threshold.

The algorithm works as follows:

-   -   1. Create a large training set of L random matrices H_(l).[46]     -   2. For each random matrix Ha, perform singular valve         decomposition to obtain V_(l) as in (3).     -   3 Align orientation of each vector in V_(l) to lie within the         same 2nr- dimensional hemisphere.     -   4. Set iteration counter i=0. Create a set of 2^(N) ^(v) random         matrices Ĥ(n).     -   5. For each matrix Ĥ(n), calculate corresponding {circumflex         over (V)}^((i))(n) using singular value decomposition.     -   6. Align orientation of each vector in {circumflex over         (V)}^((i))(n) to lie within the same 2n_(T)-dimensional         hemisphere.     -   7. For each training element H_(l) and codebook entry         {circumflex over (V)}^((i))(n). calculate the metric in (4). For         every l, choose the index n_(opt)(l) corresponding to the lowest         value of γv(n;H_(l)). Calculate the subsets L(n)={l:         n_(opt)(l)=n} for all n.     -   8. Calculate new matrix {circumflex over (V)}^((i+1))(n) as a         constrained spherical average V _(l) ^(O) of all entries V_(l)         corresponding to the same index n

{circumflex over (V)}^((i+1))(n)=V _(l) ^(O)|_(l∈L(n))   (26)

-   -   9. For each region n, where cardinality |L(n)|≠0, calculate the         mean eigenmode match matrix

$\begin{matrix} {{{\hat{E}}^{({i + 1})}(n)} = {\frac{1}{❘{L(n)}❘}{\sum\limits_{l \in {L(n)}}{V_{l}^{H}{{{\hat{V}}^{({i + 1})}(n)}.}}}}} & (27) \end{matrix}$

-   -   10. Calculate the average distortion metric

γ _(v) ^((i+1))=1/L Σ_(l)γv(n_(opt)(l); H_(l))

-   -   11. If the distortion metric fulfills |γ _(v) ^((i+1))−γ _(v)         ^((i))|/γ _(v) ^((i))<Θ, where Θ is         -   a design parameter, stop. Otherwise increase i by 1, and go             to 7).

Upon completion of the above algorithm, the final set of vectors {circumflex over (V)} can be used to calculate the regions V_(l) in (5).

The design of the transmitter modulation matrices presented in section IX.C can be modified as follows: we propose the following algorithm to optimize the set of matrices {circumflex over (B)}(i₁, i₂. . . i_(nT)):

-   -   1. Creste a large .set of Ln_(T) random matrices H_(l), where L         is the number of training sets with ny users each.     -   2. For each randon matrix H_(l), perform singular value         decomposition[68] and obtain h_(i)[70] as in (15).     -   3. Align orientation of each vector hy to lie within the same         2n_(T)-dimensional hemisphere.     -   4. For each vector ha, store[74] the index i of the         corresponding entry {circumflex over (v)}(i_(l)).     -   5. Divide[76] the entire set of matrices H_(l) into L sets with         n_(T) elements cach.     -   6. Sort[78] the indices i_(l) within each set in the ascending         order. Map[78] all unique sets of sorted eigenmode indices i_(l)         to a set of unique modulation matrix indices Ig (for example, if         n_(T)=2: (1,1) →l_(B)=1; (1,2)→l_(B)=2; (2,2)→l_(B)=3 . . . ).     -   7. In each set L(I_(B))={l:(i₁, i₂. . . . i_(nT))→I_(B)},         reorder the channel vectors b; according to the indices i_(l)         and calculate[80] the optimum B_(l) using the method from         Section VII.C.     -   8. Calculate[84] the set {circumflex over (B)}(l_(B)) as a         column-wise spherical average of all entries B; corresponding to         the same[82] index I_(B) as

∀_(n=,1. . . . nT)[{circumflex over (B)}(l_(B))]_(n)=[B_(l)]_(n)|_(L(I) _(B))   (28)

After completion of the above algorithm, the transmitter will have the set of |I_(B)| modulation matrices {circumflex over (B)}(l_(B)) corresponding to all sorted combinations of the channel eigenmode indices that can be reported by the receivers.

To clarify our notation for spherical average used in (26) and (28), we outline a method to calculate a spherical average of a set of unit-length vectors, and a spherical average of a set of unitary matrices, preserving the mutual perpendicularity of the component vectors. We use the notation v _(l)|_(l∈L) to represent a spherical average of all unit-length vectors vacontained v_(l) a set L. Based on Statistical Analysis of Spherical Data by Fisher et al., we define the spherical average as

$\begin{matrix} {{{\overset{\_}{v}}_{l}^{O}❘_{i \in L}} = {\min\limits_{x}{\sum\limits_{l \in L}{\cos^{- 1}\left( {v_{l} \cdot x} \right)}}}} & (29) \end{matrix}$

where the unit-length vector x is found using one of the constrained non-linear optimization algorithms.

In case of the spherical average of a set of unitary matrices, denoted as v _(l)|_(l∈L), the averaging of the unit-length column vectors must be performed in a way that the resulting matrix is also unitary. We represent the spherical matrix average as a collection of unit-length vectors x_(l) as V=[x₁, x₂, x₃, . . . ] and jointly optimize them as

$\begin{matrix} \left\{ \begin{matrix} {{x_{k} = {\min\limits_{x}{\sum}_{l \in L}{\cos^{- 1}\left( {\left\lbrack V_{l} \right\rbrack_{\cdot {,k}} \cdot x} \right)}}},{k = 1},2,{3\ldots}} \\ {{{x_{k} \cdot x_{l}} = 0},{k \neq l}} \end{matrix} \right. & (30) \end{matrix}$

Iunmaterial modifications may be made to the embodiments described bere without departing from what is covered by the claims.

While illustrative embodiments have been illustrated and described, it will be appreciated that various changes can be made therein without departing from the spirit and scope of the invention. 

What is claimed is:
 1. A method for use by a receiving device, the method comprising: receiving a signal and determining a precoding matrix associated with a codebook for at least one subcarrier; transmitting a first value and a second value, wherein the determined precoding matrix is determinable by using both the first value and the second value; and receiving a precoded signal using a plurality of antennas based on the first value and second value.
 2. The method of claim 1, wherein the first value is a coarse value and the second value refines the coarse value for the at least one subcarrier.
 3. The method of claim 1, wherein the precoding matrix is for a plurality of adjacent subcarriers.
 4. The method of claim 1, wherein the receiving device has a plurality of codebooks for use in determining the precoding matrix.
 5. The method of claim 1, wherein the precoded matrix is selected so that another user does not suffer from multi-user interference from the precoded signal.
 6. The method of claim 1, wherein the precoded signal is a multi-user multiple input multiple output (MIMO) signal.
 7. The method of claim 1, wherein the first value and the second value are transmitted as channel state information (CSI).
 8. A receiving device, configured for wireless reception, the receiving device comprising: receiver circuitry and processor circuitry configured to receive a signal and determine a precoding matrix associated with a codebook for at least one subcarrier; transmitter circuitry configured to transmit a first value and a second value, wherein the precoding matrix is determinable by using both the first value and the second value; and the receiver circuitry further configured to receive a precoded signal using a plurality of antennas based on the first value and second value.
 9. The receiving device of claim 8, wherein the first value is a coarse value and the second value refines the coarse value for the at least one subcarrier.
 10. The receiving device of claim 8, wherein the precoding matrix is for a plurality of adjacent subcarriers.
 11. The receiving device of claim 8, wherein the receiving device has a plurality of codebooks for use in determining the precoding matrix.
 12. The receiving device of claim 8, wherein the precoded matrix is selected so that another user does not suffer from multi-user interference from the precoded signal.
 13. The receiving device of claim 8, wherein the precoded signal is a multi-user multiple input multiple output (MIMO) signal.
 14. The receiving device of claim 8, wherein the first value and the second value are transmitted as channel state information (CSI).
 15. A transmitting device, configured for wireless transmitting, the transmitting device comprising: transmitter circuitry configured to transmit a signal for determining a precoding matrix associated with a codebook for at least one subcarrier; receiver circuitry configured to receive a first value and a second value, wherein the precoding matrix is determinable by using both the first value and the second value; and the transmitter circuitry further configured to transmit a precoded signal using a plurality of antennas based on the first value and second value.
 16. The transmitting device of claim 15, wherein the first value is a coarse value and the second value refines the coarse value for the at least one subcarrier.
 17. The transmitting device of claim 15, wherein the precoding matrix is for a plurality of adjacent subcarriers.
 18. The transmitting device of claim 15, wherein the precoded matrix is selected so that another user does not suffer from multi-user interference from the precoded signal.
 19. The transmitting device of claim 15, wherein the precoded signal is a multi-user multiple input multiple output (MIMO) signal.
 20. The transmitting device of claim 15, wherein the first value and the second value are received as channel state information (CSI). 